Maximal subgroups of free idempotent-generated semigroups factored out by biorder relations

David Easdown, Sean Gardiner and Brett McElwee

Abstract

Given any biordered set $E$, we may form the idempotent-generated semigroup $F_E$, which is generated by the set $E$, subject to the relations $ef=e\ast f$ whenever $e$ and $f$ are elements of $E$ and $e\ast f$ is a basic product. Easdown proved in 1985 that the biordered set of $F_E$ is biorder isomorphic to $E$, thus demonstrating that the biordered set axioms, introduced by Nambooripad in 1974, characterise certain partial algebras of idempotents of semigroups. Relatively little is known about the general structure of $F_E$, though it is known that every group can arise as a maximal subgroup of $F_E$ for some $E$, and that, as a consequence, the word problem is unsolvable. In this article, presentations for maximal subgroups are studied using homomorphic images of fundamental groups of graphs associated with $\mathcal{D}$-classes of the biordered set $E$. To illustrate the technique, small biordered sets $E$ are constructed where $F_E$ contains maximal subgroups which are cyclic of order two and free abelian on two generators respectively, the second of which reconstructs an example due to Dolinka.

Keywords: free idempotent-generated semigroups, biordered sets, maximal subgroups.

AMS Subject Classification: Primary 06A99; secondary 20M10, 20M20.